Data adaptive compression and data encryption using kronecker products

ABSTRACT

Digital files are compressed using a process including Schmidt decompositions of matrices using an algorithm, termed ‘BSD’ herein, which is based on an algebraic method generalizing QR decomposition. Software analyzes an input file and initially identifies a matrix M, with entries within a predefined set of integers, within the file. Next, essential entries are defined, extracted from M, that contain sufficient information to recover M using BSD. The compressed file includes the essential entries and their positions within M. To achieve an encryption process, software encrypts the pattern matrix that includes the positions of the essential entries of M. To achieve a lossy compression, software identifies essential entries that contain sufficient information to recover an approximation to M for which the quality is determined by an error threshold. For a more efficient lossy compression, software uses singular value decomposition, BSD, and other signal processing of M.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of related U.S. Patent Application No. 62/040,674, filed Aug. 22, 2014, the contents of which are incorporated herein by reference in their entirety.

FIELD OF THE DISCLOSURE

The disclosure relates to a system and method for reducing binary file size, and in particular, to file compression using matrices.

BACKGROUND OF THE DISCLOSURE

Singular Value Decomposition (SVD) yields a method to decompose each mn×pq matrix M into a sum with a minimum number of terms, each of which is the Kronecker product of an m×p matrix by and n×q matrix. This decomposition is known as a Schmidt decomposition of M. We shall say that M is decomposed with respect to the decomposition shape (m, n, p, q). Assuming that M represents a digital file, dropping some terms from the decomposition of M and using the other terms to build a matrix that approximates M leads to a lossy compression of the digital file. In addition to this compression method, there is another compression method based on SVD known as compression with SVD. Every compression method based on SVD has an energy-compaction property which causes the method to be useful for compression. With SVD, singular values and vectors are to be stored to construct the output file. These values and entries are not necessarily integers even if all entries in the original matrix are integers. Thus, storing in a computer the singular values and vectors without losing too much information requires per pixel a much larger amount of memory space than the amount of memory space occupied by a pixel in the original file. Therefore, the compression ratio with SVD is not as desirable in comparison with ratios achieved by other existing compression methods, such as JPEG [see references 4-6, below]. Other compressions schemes based on algebra include algorithms based on QR and QLP decompositions, [1A].

SUMMARY OF THE DISCLOSURE

In an embodiment of the disclosure, a method for encoding digital data comprises using a computer executing software stored on non-transitory media, the software configured to identify an mn×pq matrix M, with entries within a predefined set of integers, within the digital data; define a left essential matrix A; define a right essential matrix B; define a pattern matrix P for storing positions of essential entries; assign to matrix M_(e) a starting value of M; define a matrix A_(e); define a matrix B_(e); assign a starting value to e; a) select a non-zero entry d_(e) of M_(e); b) store the position (r, c) of the selected non-zero entry of M_(e) at an e^(th) column of P; c) select from M_(e) two matrices A_(e) and B_(e) having d_(e) as a common entry and for which A_(e)⊗B_(e)/d_(e) is a term in a BSD of M with respect to the parameters m, n, p, and q; d) store in the e^(th) m×p block of A the entries of M whose positions are the positions of the entries of A_(e) in M_(e); e) store in the e^(th) n×q block of B the entries of M whose positions are the positions of the entries of B_(e) in M_(e); and f) calculate the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e), and if a predetermined error threshold between M and M_(e+1) is reached, repeat steps (a)-(f) with M_(e+1), otherwise, P, A, and B collectively represent encoded digital data corresponding to M; and g) transfer the encoded data to digital storage on at least one of the computer executing software or another computer, the encoded data comprising fewer data bytes than the source digital data, and representing at least one of all the information in the source digital data, and an approximation of all of the information in the source digital data.

In a variation thereof, the non-zero number selected in step (a) corresponds to the first entry, with respect to the stereographic order, whose absolute value is the maximum of the absolute values of the entries of Me.

In a further variation thereof, the software is configured, in step (c), to select from M_(e) two matrices A_(e) and B_(e) by i) calculating the Euclidean division of c by q and find the remainder j, and if remainder j is zero, replace j with q, then compute i=(c−j)/q+1; and ii) calculating the Euclidean division of r by n and find the remainder 1, and if remainder 1 is zero, replace 1 with n, then compute k=(r−1)/n+1.

In a yet further variation thereof, the software is further configured to (i) carry out step (d) by, for each integer a between 1 and m, and each integer b between 1 and p, storing the entry of M at position (1+(a−1)n,j+(b−1)q) in left essential matrix A at position (a,(e−1)p+b), and storing the entry of Me at position (1+(a−1)n,j+(b−1)q) in matrix Ae at position (a, b); and (ii) carry out step (e) by, for each integer a between 1 and n, and each integer b between 1 and q, storing the entry of M at position (1+(k−1)n,b+(i−1)q) in right essential matrix B at position (a,(e−1)q+b), and storing the entry of Me at position (1+(k−1)n,b+(i−1)q) in matrix Be at position (a, b).

In other variations thereof, the predetermined stop value is infinity for a lossless encoding of the digital data; the predetermined stop value is a positive number for a lossy encoding of the digital data; the sum of the stored sizes of P, A, and B is less than the stored size of the digital data corresponding to M; and/or all of the entries of A and B are extracted from M and the entries of P are integers.

In another variation thereof, the software is further configured to use P, A, and B to decode the digital data as matrix N, the software being thus further configured to define R as the number of columns in P; define E as an mn×pq matrix; define m as the number of rows in A; define p as the number of columns in A divided by R; define n as the number of rows of B; define q as the number of columns in B divided by R; assign a starting value to e; extract an ordered pair (r, c) from the e^(th) column of P; carry out steps (c)-(f) in reverse, as steps (f)-(c), placing the entries of A and B in E at the same positions they occupy in M; fill with zero all values of E which have not been assigned an entry from A or B; assign to matrix E_(e) a starting value of E; assign a starting value to e; a) extract an ordered pair (r, c) from the e^(th) column of P; b) select d_(e) the entry of E_(e) at the position (r, c); c) using the method of steps (c)-(f) of claim 1, recover the matrices A_(e) and B_(e) of claim 1; and d) if e<R, compute E_(e+1)=E_(e)−A_(e)⊗B_(e)/d_(e) and repeat (a)-(d) with E_(e+1), and if e=R, compute N=A₁⊗B₁+A₂⊗B₂+. . . +A_(R)⊗B_(R).

In other variations thereof, in step (d), if N=M, the digital data has been decoded without a loss of data, and if N≠M, N approximates the digital data; and/or the digital data corresponding to M is encrypted by encrypting P.

In yet another variation thereof, the software is further configured to use P, A, and B to decode the digital data as a matrix that approximates M; wherein M is one of a plurality of matrices M identified within the digital data, and wherein steps (a)-(f) are performed for each of the plurality of matrices M, and wherein the digital data is encoded by the collective matrices of P, A, and B corresponding to the plurality of matrices M along with the size S of the matrix M_(E) that represents the entire digital data.

In another embodiment of the disclosure, a method for encoding digital data, comprises using a computer executing software stored on non-transitory media, the software configured to define R as the number of columns in P; define E as an mn×pq matrix; define m as the number of rows in A; define p as the number of columns in A divided by R; define n as the number of rows of B; define q as the number of columns in B divided by R; assign a starting value to e; extract an ordered pair (r, c) from the e^(th) column of P; select d_(e) the entry of E_(e) at the position (r, c); calculate the Euclidean division of c by q and find the remainder j, and if remainder j is zero, replace j with q, then compute i=(c−j)/q+1; calculate the Euclidean division of r by n and find the remainder l, and if remainder l is zero, replace l with n, then compute k=(r−l)/n+1; for each integer a between 1 and mn, and each integer b between 1 and pq, copy the entry of A at position (a,(e−1)p+b) into position (l+(a−1)n,j+(b−1)q) of E, and copy the entry of B at position (a,(e−1)q+b) into position (a+(k−1)n,b+(i−1)q) of E; fill with zero all values of E which have not been assigned an entry from A or B; assign to matrix E_(e) a starting value of E; assign a starting value to e; a) extract an ordered pair (r, c) from the e^(th) column of P; b) select d_(e) the entry of E_(e) at the position (r, c); c) calculate the Euclidean division of c by q and find the remainder j, and if remainder j is zero, replace j with q, then compute i=(c−j)/q+1; d) calculate the Euclidean division of r by n and find the remainder l, and if remainder l is zero, replace l with n, then compute k=(r−l)/n+1; e) for each integer a between 1 and m, and each integer b between 1 and p, store the entry of E_(e) at the position (l+(a−1)n,j+(b−1)q) at the position (a, b) of matrix A_(e); f) for each integer a between 1 and n, and each integer b between 1 and q, store the entry of E_(e) at the position (l+(k−1)n,b+(i−1)q) at the position (a, b) of matrix B_(e); and g) if e<R, compute E_(e+1)=E_(e)−A_(e)⊗B_(e)/d_(e) and repeat (a)-(g) with E_(e+1), and if e=R, compute N=A₁⊗B₁+A₂⊗B₂+. . . +A_(R)⊗B_(R); and h) transfer the encoded data to digital storage on at least one of the computer executing software or another computer, the encoded data comprising fewer data bytes than the source digital data, and representing at least one of all the information in the source digital data, and an approximation of all of the information in the source digital data.

In a variation thereof, the software is further configured to use P, A, and B to decode the digital data as a matrix that approximates M; wherein M is one of a plurality of matrices M identified within the digital data, and wherein steps (a)-(f) are performed for each of the plurality of matrices M, and wherein the digital data is encoded by the collective matrices of P, A, and B corresponding to the plurality of matrices M along with the size S of the matrix M_(E) that represents the entire digital data.

In another variation thereof, each matrix M has a predetermined size, and wherein if the digital data is not evenly divisible by the predetermined size, a remaining partial matrix M_(p) is padded with zeros to the predetermined size, and are discarded when the digital data is decoded.

In a further embodiment of the disclosure, a method for encoding digital data, comprises using a computer executing software stored on non-transitory media, the software configured to use SVD to find Ma, an mn×pq matrix, with the lowest Schmidt rank R for which PSNR(Ma, M)≥a predetermined value; quantize Ma to find a matrix M whose entries are integers; define a left essential matrix A; define a right essential matrix B; define a pattern matrix P for storing positions of essential entries; assign to matrix M_(e) a starting value of M; define a matrix A_(e); define a matrix B_(e); assign a starting value to e; a) select a non-zero entry d_(e) of M_(e); b) store the position (r, c) of the selected non-zero entry of M_(e) at an e^(th) column of P; c) select from M_(e) two matrices A_(e) and B_(e) having d_(e) as a common entry and for which A_(e)⊗B_(e)/d_(e) is a term in the Schmidt decomposition of M with respect to the parameters m, n, p, and q; d) store in the e^(th) m×p block of A the entries of M whose positions are the positions of the entries of A_(e) in M_(e); e) store in the e^(th) n×q block of B the entries of M whose positions are the positions of the entries of B_(e) in M_(e); f) calculate the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e), and if e<R, repeat steps (a)-(f) with M_(e+1), and if e=R, then P, A, and B collectively represent encoded digital data corresponding to M.

In another embodiment of the disclosure, a method for encoding digital data, comprises using a computer executing software stored on non-transitory media, the software configured to: identify an mn×pq matrix M, with entries within a predefined set of integers, within the digital data; define an essential sequence S which is in two part S₁ and S₂; define an essential matrix E; define a pattern sequence PS for storing positions of n×q block matrices of E; define a matrix A_(e); define a matrix B_(e); assign to matrix M_(e) a starting value of M and assign a starting value to e; a) select a non-zero entry d_(e) of M_(e); b) store at the e^(th) term of PS the position of the e^(th) n×q block of M_(e) that includes (r, c) the position of d_(e); c) store the e^(th) n×q block matrix of M at the e^(th) n×q block matrix of E, and following the lexicographic order, store the entries of the e^(th) n×q block of M in S₁; d) calculate the Euclidean division of c by q and find the remainder j, and if the remainder is zero, replace j with q, then compute i=(c−j)/q+1; and calculate the Euclidean division of r by n and find the remainder l, and if the remainder is zero, replace l with n, then compute k=(r−l)/n+1; e) for each integer a between 1 and m and each integer b between 1 and p, store at position (a, b) of matrix A_(e) the entry of M_(e) at position (l+(a−1)n,j+(b−1)q), and following the lexicographic order, if position (l+(a−1)n,j+(b−1)q) of E is unfilled, store in it the entry of M located at the same position and store the same entry in sequence S₂; and f) calculate the matrix M_(e+1)=M_(e) 31 A_(e)⊗B_(e)/d_(e), and if a predetermined error threshold between M and M_(e+1) is reached, repeat steps (a)-(f) with M_(e+1), otherwise, collect S₁ and S₂ to form a sequence S, which with PS and the shape (m, n, p, q) represent encoded digital data corresponding to M; and g) transfer the encoded data to digital storage on at least one of the computer executing software or another computer, the encoded data comprising fewer data bytes than the source digital data, and representing at least one of all the information in the source digital data, and an approximation of all of the information in the source digital data.

In a variation thereof, the software is configured to de-encode the encoded digital data, by being further configured to: define R as the number the number of terms in PS; define E as an mn×pq matrix; use the first Rnq terms of S to build R n×q block matrices of E, use PS to identify the positions of the R block matrices, and delete from S the used terms, then call essential position any filled position of E; assign to matrix M_(e) a starting value of E, and assign a starting value to e; a) build a matrix B_(e) whose entries are the entries of the e^(th) n×q block matrix of M_(e), and following the lexicographic order compute (r, c) the position in M_(e) of the first entry d_(e) in B_(e) whose absolute value is equal to the maximum value of the absolute values of the entries of B_(e), then store d_(e) at the e^(th) term of a sequence D; b) calculate the Euclidean division of c by q and find the remainder j, and if the remainder is zero, then replace j with q, then compute i=(c−j)/q+1, and calculate the Euclidean division of r by n and find the remainder l, and if the remainder is zero, replace l with n, then compute k=(r−l)n+1, and store i the e^(th) term of a sequence I, j the e^(th) term of a sequence J, k the e^(th) term of a sequence K, and l the e^(th) term of a sequence L; c) build an an m×p matrix A_(e) as follows, for each (a, b), where 1≤a≤m and 1≤b≤p, if (l+(a−1)n,j+(b−1)q) is an essential position, then fill position (a, b) of A_(e) with the entry of M_(e) located at position (l+(a−1)n,j+(b−1)q), fill this position of E with the first term of S and mark (l+(a−1)n,j+(b−1)q) as an essential position of E; d) fill with zero the other positions of A_(e); f) compute M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e), and if e<R, then repeat steps (a)-(d) with M_(e+1), and if e=R, then assign to matrix E_(e) a starting value of E, and assign a starting value to e; g) compute i, j, k, and l that are, respectively, the e^(th) term of I, the e^(th) term J, the e^(th) term of K, and the e^(th) term of L; h) build an m×p matrix A_(e) and an n×q matrix B_(e) are built as follows, for each integer a between 1 and m and each integer b between 1 and p, place at position (a, b) of A_(e) the entry of E_(e) at position (l+(a−1)n,j, and assign the e^(th) n×q block matrix of E_(e) to B_(e); k) if e<R, then compute E_(e+1)=E_(e)−A_(e)⊗B_(e)/d_(e), where d_(e) is the e^(th) term of D, and repeat steps (g) and (h) with E_(e+1), and if e=R, then compute N=A₁⊗B₁/d₁+A₂⊗B₂/d₂+. . . +A_(R)⊗B_(R)/d_(R), wherein the matrix N is the matrix of the output file.

In a further variation thereof, lossless compression is achieved by stopping iterative computation of steps (a)-(d) in step (f) after the Schmidt rank of M is reached.

In a still further variation thereof, error is measured by using at least one of PSNR, PEVQ, and SSIM.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete understanding of the present disclosure, and the attendant advantages and features thereof, will be more readily understood by reference to the following detailed description when considered in conjunction with the accompanying drawings wherein:

FIG. 1A is a diagram of an embodiment of BSD Lossless Compression in accordance with the disclosure, wherein the algorithms diagrammed are executed from top to bottom and left to right, and wherein letters in circles throughout the drawings refer to logical flow patterns between the various algorithms;

FIG. 1B is a diagram of an embodiment of BSD Lossless Decompression in accordance with the disclosure;

FIG. 1C is a diagram of an embodiment of BSD Lossy Compression in accordance with the disclosure;

FIG. 2A is a diagram of an embodiment of BSD Lossy Compression using SVD and blocks, in accordance with the disclosure;

FIG. 2B is a diagram of an embodiment of BSD Decompression, where SVD and blocks were used, in accordance with the disclosure;

FIG. 3A is a diagram of an embodiment of BSD Compression and Encryption in accordance with the disclosure;

FIG. 3B is a diagram of an embodiment of BSD Decompression in accordance with the disclosure; and

FIG. 4 depicts computing equipment, part or all of which can be used to carry out the

disclosure.

DETAILED DESCRIPTION OF THE DISCLOSURE

As required, detailed embodiments are disclosed herein; however, it is to be understood that the disclosed embodiments are merely examples and that the systems and methods described below can be embodied in various forms. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a basis for the claims and as a representative basis for teaching one skilled in the art to variously employ the present subject matter in virtually any appropriately detailed structure and function. Further, the terms and phrases used herein are not intended to be limiting, but rather, to provide an understandable description of the concepts.

The terms “a” or “an”, as used herein, are defined as one or more than one. The term plurality, as used herein, is defined as two or more than two. The term another, as used herein, is defined as at least a second or more. The terms “including” and “having,” as used herein, are defined as comprising (i.e., open language). The term “coupled,” as used herein, is defined as “connected,” although not necessarily directly, and not necessarily mechanically. Herein, reference to ‘the software’ refers to electronic execution of instructions embodying the methods of the disclosure, the methods of the disclosure thereby being carried out by one or more electronic processors, including, for example, parallel processors. The term ‘memory value’, or size in memory, may be considered to also indicate the size of the referenced element within non-volatile storage.

In an embodiment of the disclosure, digital files are compressed using a method which includes Schmidt decompositions of matrices. The disclosure provides for performing lossless and lossy compressions of any digital file that can be represented by a matrix M. The lossless compression is achieved by an apparatus that stores some entries of M to recover M. The lossy compression is achieved by an apparatus that combines methods of the disclosure with other techniques that include low rank approximation and compressing by blocks. In another aspect, an apparatus executes software configured to execute the methods of the disclosure to compresses a file, and in an embodiment, enables a user to choose a key code that is needed at the decompression process. Accordingly, the present invention enables data compression and data security.

The disclosure provides a method to find Schmidt decompositions of matrices. Herein, this method will be referred to as BSD (Bourouihiya Schmidt Decomposition). A BSD of a matrix M uses the entries of M and elementary operations to compute each entry of each matrix involved in the decomposition of M. The disclosure uses this fact to enable lossless and lossy compressions of digital files. In an embodiment, a lossless compression is almost impossible with SVD, however by combining BSD and SVD, the invention enables lossy compression while solving the storage problems encountered by compressing with SVD, and with a very low cost in term of error. Compression in accordance with the disclosure is competitive with popular existing methods, such as JPEG or JPEG2000.

The present disclosure provides the following embodiments: the use of BSD to achieve lossless or lossy compressions of digital files; the use of BSD, low rank approximation with SVD, compressing by blocks, and other techniques to achieve efficient lossy compressions of digital files, including enabling a user to choose from a multitude of parameters to adapt the compression method to a specific application; and encryption of a digital file, wherein a user can choose a key code that is required for decryption at the decompression process, wherein the encryption methods of the disclosure can be used with the lossy and lossless compressions of the disclosure.

Lossless compression of digital data or files is accomplished using BSD, as follows, although this method may also be used to achieve lossy compression:

Each mn×pq-matrix M can be written as

M=A ₁ ⊗B ₁ +A ₂ ⊗B ₂ +. . . +A _(R) ⊗B _(R),   (4.1)

where A_(k) is an m×p-matrix and B_(k) is an n×q-matrix. In the art, Equality (4.1) is known as a Schmidt decomposition of M and R is the (Schmidt) rank if the number of terms in (4.1) is the minimum possible. We shall denote rank (M)=R. If M is decomposed using BSD, then each entry in A_(k) or B_(k) is computed using entries extracted from M and the elementary operations, for each k=1, . . . ,R.

Example 1

Consider the 6×6-matrix M whose entries are integers between 0 and 255. Thus, 36 Bytes is the memory space required to store M in a computer. In this example, BSD is used to decompose M with respect to (3, 2, 2, 3), i.e., into a sum of matrices, for which each term is the Kronecker product of a 3×2 matrix by a 2×3 matrix.

$\begin{matrix} {{M = \begin{pmatrix} \underset{\_}{1} & \underset{\_}{2} & \underset{\_}{3} & \underset{\_}{4} & 5 & 6 \\ \underset{\_}{7} & \underset{\_}{8} & 9 & 10 & 11 & \underset{\_}{12} \\ \underset{\_}{13} & 14 & 15 & \underset{\_}{16} & 17 & 18 \\ 19 & 20 & \underset{\_}{21} & 22 & 23 & \underset{\_}{24} \\ \underset{\_}{25} & 26 & 27 & \underset{\_}{28} & \underset{\_}{29} & \underset{\_}{30} \\ 31 & 32 & \underset{\_}{33} & \underset{\_}{34} & \underset{\_}{35} & \underset{\_}{36} \end{pmatrix}}{{M = {{A_{1} \otimes B_{1}} + {A_{2} \otimes B_{2}}}},}} & (4.2) \end{matrix}$

where

$A_{1} = \begin{pmatrix} 9 & 12 \\ 21 & 24 \\ 33 & 36 \end{pmatrix}$ $B_{1} = \begin{pmatrix} {28\text{/}36} & {29\text{/}36} & {30\text{/}36} \\ {34\text{/}36} & {35\text{/}36} & {36\text{/}36} \end{pmatrix}$ $A_{2} = \begin{pmatrix} {1 - {9\left( {28\text{/}36} \right)}} & {4 - {12\left( {28\text{/}36} \right)}} \\ {13 - {21\left( {28\text{/}36} \right)}} & {16 - {24\left( {28\text{/}36} \right)}} \\ {25 - {33\left( {28\text{/}36} \right)}} & {28 - {36\left( {28\text{/}36} \right)}} \end{pmatrix}$ $B_{2} = \begin{pmatrix} \frac{1 - {9\left( {28\text{/}36} \right)}}{1 - {9\left( {28\text{/}36} \right)}} & \frac{2 - {9\left( {29\text{/}36} \right)}}{1 - {9\left( {28\text{/}36} \right)}} & \frac{3 - {9\left( {30\text{/}36} \right)}}{1 - {9\left( {28\text{/}36} \right)}} \\ \frac{7 - {9\left( {34\text{/}36} \right)}}{1 - {9\left( {28\text{/}36} \right)}} & \frac{8 - {9\left( {35\text{/}36} \right)}}{1 - {9\left( {28\text{/}36} \right)}} & \frac{9 - {9\left( {36\text{/}36} \right)}}{1 - {9\left( {28\text{/}36} \right)}} \end{pmatrix}$

The entries of A₁, A₂, B₁, and B₂ are computed using the essential sequence S=(9, 12, 21, 24, 33, 36, 28, 29, 30, 34, 35, 1, 4, 13, 16, 25, 2, 3, 7, 8) whose terms are called essential entries of M (these entries are bolded and underlined in M). The methodology for selecting essential entries is provided elsewhere herein. To recover M, we need to store the decomposition shape (3, 2, 2, 3), the essential sequence S, and a pattern matrix that includes the position (6, 6) (used to locate A₁ and B₁) and the position (1, 1) (used to locate A₂ and B₂). Since there are 20 essential entries, the decomposition shape includes 4 entries, and the pattern matrix includes 4 entries, then 28 Bytes is the memory space required to store the compressed file. Thus, a lossless compression of M is achieved, from 36 Bytes to 28 Bytes. A mathematical proof for the BSD algorithm is provided elsewhere herein.

If the Schmidt decomposition of M was carried out using SVD, then

M = N₁₁ ⊗ N₂₁ + N₁₂ ⊗ N₂₂,

where,

${N_{11} \simeq \begin{pmatrix} {- 13.1341} & {- 20.4201} \\ {- 42.2782} & {- 49.5642} \\ {- 71.4223} & {- 78.7083} \end{pmatrix}},{N_{21} \simeq \begin{pmatrix} {- 0.3365} & {- 0.3535} & {- 0.3706} \\ {- 0.4389} & {- 0.4560} & {- 0.4731} \end{pmatrix}},{N_{12} \simeq {\begin{pmatrix} {- 5.9578} & {- 5.0018} \\ {- 2.1338} & {- 1.1778} \\ 1.6901 & 2.6461 \end{pmatrix}\mspace{14mu} {and}\mspace{14mu} N_{22}} \simeq {\begin{pmatrix} 0.5739 & 0.4437 & 0.3135 \\ {- 0.2073} & {- 0.3375} & {- 0.4677} \end{pmatrix}.}}$

Every entry in the matrice N₁₁, N₂₁, N₁₂, or N₂₂ is not an integer between 0 and 255 and can only be computed approximately. Therefore, storing these matrices does not achieve a lossless compression of M. This is because, storage of real numbers greatly increases the size of the compressed file.

Using BSD for image compression, it is unlikely that storing the compressed file requires a memory space that is more than the memory space required to store the original file. As a practical matter, most compressions are substantially smaller than the original file using the methods of the disclosure.

Embodiment 1

With reference to FIG. 1A, in an embodiment of the disclosure, lossless compression of digital files is achieved with BSD. As explained elsewhere herein, lossy compression can be achieved with a similar method.

Assume that the input file is represented by an mn×pq matrix M that will be decomposed with respect to the decomposition shape (m, n, p, q). In the course of the compression and decompression processes, we shall define and compute the following matrices and sequence.

P: The pattern matrix of M.

E: The essential matrix of M, this is an mn×pq matrix.

S: Essential sequence, built using the pigeonhole principle (it should be understood that using the pigeonhole principle is optional, as detailed elsewhere, herein).

With reference to FIG. 1A, the following sets of steps describe a BSD of M. In the following, we describe the e^(th) set where e is an integer between 1 and the rank of M that will be computed during the compression process. At this set of steps, an mn×pq matrix M_(e) is defined. At 100, the compression process starts with M₁=M.

Step 1: The software computes d, the maximum value of the absolute values of the entries of M_(e). Following the lexicographic order, position (r, c) in M_(e) of the first entry d_(e) whose absolute value is equal to d is stored at the e^(th) column of the pattern matrix P (102).

Step 2: The software performs the Euclidean division of c by q and find the reminder j. If the reminder is zero, then j is replaced with q. The software then computes i=(c−j)/q+1.

Step 3: The software performs the Euclidean division of r by n and find the reminder l. If the reminder is zero, then l is replaced with n. The software then computes k=(r−l)/n+1.

Step 4: For each integer a between 1 and m and each integer b between 1 and p, the entry of M_(e) at position (l+(a−1)n,j+(b−1)q) is stored at position (a, b) of the matrix A_(e) (104). Following the lexicographic order, if position (l+(a−1)n,j+(b−1)q) of E is unfilled, we store in it the entry of M located at the same position; and we store the same entry in the sequence S (106). This entry is then called an essential entry of M.

Step 5: For each integer a between 1 and n and each integer b between 1 and q, the entry of M_(e) at position (a+(k−1)n,b+(i−1)q) is stored at position (a, b) of the matrix B_(e) (108). Following the lexicographic order, if position (a+(k−1)n,b+(i−1)q) of E is unfilled, we store in it the entry of M located at the same position (106); and we store the same entry in the sequence S (108). This entry is then called an essential entry of M.

Step 6: The software computes the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e). (110) (As detailed elsewhere herein, the disclosure proves the fact that rank (M_(e)−A_(e)⊗B_(e)/d_(e))=rank(M_(e))−1)

If M_(e+1)=0, then the lossless compression of M is achieved (112).

If M_(e+1)≠0, then Steps 1-6 are performed with M_(e+1) (114).

The process stops after R sets of steps, where R is the Schmidt rank of M.

With reference to FIG. 1C, the lossy compression process includes sets of the same steps 1-5, while an error threshold is chosen at Step 1 (140); and the second statement of Step 6 is replaced with a statement that stops the process when the error threshold is exceeded (142).

The decomposition shape (m, n, p, q), the pattern matrix P, and the essential sequence S constitute the compressed file.

There are other possible embodiments of BSD, including for example replacing step 1 with “Following the lexicographic order, position (r, c) of the first nonzero entry d_(e) of M_(e) is stored at the e^(th) column of the pattern matrix P.” Various embodiments can lead to different compression results.

The lossless compression described in this embodiment assumes that the size of the data matrix fits with the shape of the Schmidt decomposition. In other cases, the data matrix is augmented to fit the shape and the added entries are disregarded at the decompression process. Notice, that there are shapes (such as (m, n, 1, 1)) that fit with the size of every matrix.

Embodiment 1 Decompression

With reference to FIG. 1B, in the decompression process, the quadruple (m, n, p, q), the matrix P, and the sequence S constitute the input (120).

Step 1: The software computes R that is equal to the number of columns in P (122).

Step 2: Starting with e=1 and finishing with e=R, the apparatus performs the following substeps for each e (124).

Step 2.1: The apparatus extracts the ordered pair (r, c) from the e^(th) column of P.

Step 2.2: The software performs the Euclidean division of c by q and find the reminder j. If the reminder is zero, then j is replaced with q. The software then computes i=(c−j)/q+1.

Step 2.3: The software performs the Euclidean division of r by n and find the reminder l. If the reminder is zero, then l is replaced with n. The software then computes k=(r−l)/n+1.

Step 2.4. Following the lexicographic order of the set {(a, b): 1≤a≤m and 1≤b≤p}, the apparatus stores the first terms of S that are enough to fill in E every unfilled position (l+(a−1)n,j+(b−1)q); S is then replaced with a new sequence that includes all terms of S but the terms used in Step 2.4 for e. After that, the old sequence is deleted and the new sequence is called S.

Step 2.5. Following the lexicographic order of the set {(a, b): 1≤a≤n and 1≤b≤q}, the apparatus stores the first terms of S that are enough to fill in E every unfilled position (a+(k−1)n,b+(i−1)q); S is then replaced with a new sequence that includes all terms of S but the terms used in Step 2.5 for e. After that, the old sequence is deleted and the new sequence is called S.

Step 3: The software fills with 0 each position in E that is not filled after step 2 (126).

Step 4: At this step, an mn×pq matrix E_(e) is defined, for each e=1, . . . ,R. The process starts with E₁=E. The software extracts the ordered pair (r, c) from the e^(th) column of P, extracts the entry d_(e) of E_(e) at position (r, c), and performs the following sub-steps (128).

Step 4.1: The software performs the Euclidean division of c by q and find the reminder j. If the reminder is zero, then j is replaced with q. The software then computes i=(c−j)/q+1.

Step 4.2: The software performs the Euclidean division of r by n and find the reminder l. If the reminder is zero, then l is replaced with n. The software then computes k=(r−l)/n+1.

Step 4.3: For each integer a between 1 and m and each integer b between 1 and p, the entry of E_(e) at the position (l+(a−1)n,j+(b−1)q) is a is placed at position (a, b) of A_(e).

Step 4.4: For each integer a between 1 and n and each integer b between 1 and q, the entry of E_(e) at the position (a+(k−1)n,b+(i−1)q) is placed at position (a, b) of B_(e).

If e<R, then the software computes E_(e+1)=E_(e)−A_(e)⊗B_(e)/d_(e) and redo sub-Steps 4.1-4.4 (130).

If e=R, then the software performs Step 5.

Step 5: The software collect the matrices found in Step 4 as follows:

N=A ₁ ⊗B ₁ /d ₁ +A ₂ ⊗B ₂ /d ₂ +. . . +A _(R) ⊗B _(R) /d _(R).

The matrix N is the matrix of the output file. If the compression is lossless, then N=M (132). If the compression is lossy, then N approximates M.

Assume that a digital file is represented by M whose entries are integers between 0 and 255. Thus, every pixel in the file requires 1 byte to be stored in a computer. To store the decomposition shape we need 4 bytes. The terms of the essential sequence S are extracted from M, and hence, the memory space required to store S is less than the memory space required to store M. The number of entries of P is less than the minimum of the numbers 2mp and 2nq, while M counts mnpq entries. Practically, the pattern matrix P occupies less than 1% of the memory space occupied by M. Thus, compression with BSD has a lossless compression ratio that is more than 0.99. For example, if m=16, n=24, p=8, and q=16, the memory size of the matrix M is 49,152 bytes. Meanwhile, to store the decomposition shape (16, 24, 8, 16) and the pattern matrix P, we need no more than 260 bytes, that is less than 0.53% of the size of M. Thus, the compression ratio is more than 0.994.

In many cases, there are several choices for the decomposition shape (m, n, p, q). Some of these choices are more appropriate to adapt to some type of files to be compressed. For example, to decommpose a 256×256 matrix, we can chose from 49 decomposition shapes.

The disclosure uses algebraic computations that include the Kronecker product. This can be distinguished from lossless compression methods based on statistical models and bit sequences that include Huffman coding and arithmetic coding, which are logical and statistical. The disclosure provides algebraic lossless compression that can be applied to digital data.

Embodiment 2

With reference to FIG. 2A, the second embodiment uses BSD, low rank approximation, and compressing by blocks, to achieve efficient lossy compressions of digital files. To measure the quality of the output file, Peak Signal to Noise Ratio (PSNR) is used. It should be understood that while PSNR is referenced herein as a measurement of quality or error, there are many different types of measurements of error which can be substituted, for example root means squared, PEVQ (Perceptual Evaluation of Video Quality, which is advantageous for video data), and SSIM (Structural Similarity Index Method, which, like PSNR, is advantageous for images), where the other measurement is selected as best suits the type of original data file used, or as otherwise deemed suitable for a particular purpose.

Assume that the input file is an image represented by a matrix M whose entries are integers in [0, 255], for which we want the entries of the compressed file and output file to be also integers in [0, 255]. Thus, each entry can be stored with one byte in a digital device.

For the compression process, the inputs include M, Er, a number that sets a threshold error, and the parameters m, n, p, and q that define the block matrices. We assume that the size of M is several times bigger than the size of an mn×pq-matrix. In the process of compression, M will be split into mn×pq-block matrices. In the course of the compression process the apparatus builds a pattern matrix P, an essential sequence S, and a sequence of ranks R.

Step 1: If the height of M is not divisible by mn or the width of M is not divisible by pq, then columns or rows of zeroes are added to M to reach the divisibility (160). The added parts are then disregarded at the decompressing process.

Step 2: The software splits the image into blocks. Each block is an mn×pq-matrix. Let's say that h is the number of these blocks (162). BSD will be used with respect to (m, n, p, q).

Step 3: For each e=1, . . . ,h, the software performs the following steps for the e^(th) block M_(e).

Step 3.1: Using SVD, the software computes the matrix D_(e) that has the lowest rank possible R_(e) among all matrices for which the PSNR with M is bigger or equal to Er (164).

Step 3.2: The software rounds up the matrix D_(e) and obtains a matrix Q_(e) whose entries are integers. (Most of the entries of Q_(e) are in [0, 255], but few entries maybe out of that range) (166).

Step 3.3: Using Embodiment 1, the apparatus compresses Q_(e) and computes the pattern matrix P_(e) and the essential sequence S_(e) (168).

Step 3.4: The software stores the rows of the e^(th) pattern matrix in the rows (2e−1) and 2e of the pattern matrix P; the terms of S_(e) in S (successively after the last term of S), and R_(e) in R (170).

The quadruple (m, n, p, q), the matrix P, the sequence S, the sequence R, and the size of M constitute the compressed file (172).

Embodiment 2 Decompression

With reference to FIG. 2B, for the decompression process, the input is the compressed file that includes the quadruple (m, n, p, q), the matrix P, the sequence S, the sequence R, and the size of M (180).

Step 1. The software computes h as the number of terms of R (182).

Step 2. For each e=1, . . . ,h, the apparatus performs the following steps to find the e^(th) block of the output matrix.

Step 2.1. The software extracts from the input matrices the matrix P_(e) as the rows (2e−1) and 2e of the pattern matrix P and extracts from R the number R_(e) (184).

Step 2.2. With the matrix P_(e), the number R_(e), and the sequence S, the apparatus uses steps 2-5 of the decompression process of embodiment 1 to build a block matrix N_(e) that approximates the original block M_(e) (186). The pigeonhole principle is used to replace S with a new sequence that includes all terms of S but the terms used in Step 2.2. After that the old sequence is deleted and the new sequence is called S (188).

Step 3. The software collects the h blocks built in Step 2 as one matrix N (190). The software then uses the size of M and a quantization of N to build a matrix N_(q) whose entries are integers in [0, 255], N_(q) has the size of M, and N_(q) approximates M (192).

Compressing by blocks or tiling allow parallel processing that significantly reduces the compression and decompression time.

Embodiment 3

With reference to FIG. 3A, in a third embodiment, software of the disclosure encrypts digital files, in addition to performing compression. After compressing a file, the software enables a user to choose a key code that is needed at the decompression process. The encryption software can be used with the lossy or a lossless compression methods of the disclosure.

In an example of the encryption process, assume that the input file is represented by an mn×pq-matrix M.

Step 1. The software compresses the file using the lossless compression method described in the first embodiment. The outputs are then the essential sequence S and the pattern matrix P (200).

Step 2. The user chooses a key code K, a number that can be of one to hundreds of digits (202).

Step 3. This step can be any one-to-one algorithm implemented in the software to convert the matrix P to a matrix L, using the key code (204).

The encrypted file includes the sequence S, the matrix L, the shape (m, n, p, q), and the user key code K (206).

There are numerous possible embodiments which can be used to implement an algorithm for step 3. In an example, assume that the user chose a number K=K₁K₂ . . . K_(h) with h digits and h is less or equal to the number of rows of P. For each e=1, . . . ,h, the software shifts the entries of the e^(th) row of P in a circular way by K_(e) places. Other known and hereinafter developed methods can be used to convert P to L using the key code.

Embodiment 3 Decompression/Decryption

For the decryption process, the input includes S, L, K, and (m, n, p, q) (210).

Step 1. The software uses K, L, and the inverse of the algorithm of the third step in the encryption process to recover P (212).

Step 2. The software uses the lossless decompression method, described in the first embodiment, with S, P, and (m, n, p, q) to recover M (214).

The user can use an encryption apparatus executing software of the present disclosure, to secure her/his files without the involvement of a third-party. The choice by the user of the encrypting algorithm for Step 3 in the encryption process can be a part of the key code. The decomposition shape (m, n, p, q) can be also part of the key code. All these options render access of secured data by unauthorized people more difficult.

In one embodiment, for example useful within a corporate environment, software of the disclosure can be configured to generate an encrypting algorithm for each employee. An administrator manager will have access to the encrypting algorithm of each employee, while the employee can choose a key code that locks and unlocks her/his files. If one employee stops working for the company and the company wants the former employee to become unable to access the company files, the administartor has only to change the encrypting algorithm for that employee. No employee will be required to reset her/his key code.

The lossless compression method used in the first step of the encryption process can be replaced by any lossy compression method described in the first or second embodiment.

Embodiment 4

In an embodiment of the disclosure, lossless compression of digital files is achieved with BSD. As explained elsewhere herein, lossy compression can be achieved with a similar method.

Assume that the input file is represented by an mn×pq-matrix M that will be decomposed with respect to the decomposition shape (m, n, p, q). In the course of the compression and decompression processes, we shall define and compute the following matrices.

P: The pattern matrix of M, this is a 2×R-matrix.

E: The essential matrix of M, this is an mn×pq-matrix.

A: The left essential matrix of M, this is an m×Rp-matrix.

B: The right essential matrix of M, this is an n×Rq-matrix.

The following steps describe a BSD decomposition of M. In the following, we describe the e^(th) set, where e is an integer between l and the rank of M that will be computed during the compression process. In this set of steps, an mn×pq-matrix M_(e) is defined. The compression process starts with M₁=M.

Step 1: The software computes the maximum value d of the absolute values of the entries of M_(e). Following the stereographic order, the position (r, c) of the first entry of M_(e) that is equal to d is stored at the e^(th) column of the pattern matrix P.

Step 2: The software performs the Euclidean division of c by q and find the reminder j. If the reminder is zero, then j is replaced with q. The software then computes i=(c−j)/q+1.

Step 3: The software performs the Euclidean division of r by n and find the reminder 1. If the reminder is zero, then 1 is replaced with n. The software then computes k=(r−1)/n+1.

Step 4: For each integer a between 1 and m and each integer b between 1 and p, the entry of M at the position (1+(a−1)n,j+(b−1)q) is a left essential entry. This entry is stored at the position (a,(e−1)p+b) of the left essential matrix A. The entry of M_(e) at the position (1+(a−1)n,j+(b−1)q) is stored at the position (a, b) of the matrix A_(e).

Step 5: For each integer a between 1 and n and each integer b between 1 and q, the entry of M at the position (a+(k−1)n,b+(i−1)q) is a right essential entry. This entry is stored at the position (a,(e−1)q+b) of the right essential matrix B. The entry of M_(e) at the position (a+(k−1)nb+(i−1)q) is stored at the position (a, b) of the matrix B_(e).

Step 6: The software computes the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e). (As detailed elsewhere herein, the disclosure proves the fact that rank (M_(e)−A_(e)⊗B_(e))=rank(M_(e))−1)

If M_(e+1)=0, then the lossless compression of M is achieved.

If M_(e+1)≠0, then Steps 1-6 are performed with M_(e+1).

The lossy compression process includes sets of the same steps 1-5, while the second statement of Step 6 is replaced with an error threshold that stops the process. P, A and B constitute the compressed file.

Embodiment 4 Decompression

In the decompression process, the matrices P, A and B constitute the input.

Step 1: The software computes R the number of columns in P, m the number of rows of A, p the number of columns of A divided by R, n the number of rows of B, and q the number of columns of B divided by R.

Step 2: For each e=1, . . . ,R, the apparatus extracts the ordered pair (r, c) from the e^(th) column of P and performs the following sub-steps to fill the essential matrix E with the essential entries that are left from A and right from B.

Step 2.1: The software performs the Euclidean division of c by q and find the reminder j. If the reminder is zero, then j is replaced with q. The software then computes i=(c−j)/q+1.

Step 2.2: The software performs the Euclidean division of r by n and find the reminder 1. If the reminder is zero, then 1 is replaced with n. The software then computes k=(r−1)/n+1.

Step 2.3: For each integer a between 1 and mn and each integer b between 1 and pq, the entry of A at the position (a,(e−1)p+b) is placed at the position (1+(a−1)n,j+(b−1)q) of E; and the entry of B at the position (a,(e−1)q+b) is placed at the position (a+(k−1)n,b+(i−1)q) of E.

Step 3: The software fills with 0 the positions in E corresponding to the positions of nonessential entries in M.

Step 4: At this step, an mn×pq-matrix E_(e) is defined, for each e=1 . . . R. The process starts with E₁=E. The software extracts the ordered pair (r, c) from the e^(th) column of P and performs the following sub-steps.

Step 4.1: The software performs the Euclidean division of c by q and find the reminder j. If the reminder is zero, then j is replaced with q. the software then computes i=(c−j)/q+1.

Step 4.2: The software performs the Euclidean division of r by n and find the reminder 1. If the reminder is zero, then 1 is replaced with n. The software then computes k=(r−1)/n+1.

Step 4.3: For each integer a between 1 and m and each integer b between 1 and p, the entry of E_(e) at the position (1+(a−1)n,j+(b−1)q) is a is placed at the position at (a, b) of the matrix A_(e).

Step 4.4: For each integer a between 1 and n and each integer b between 1 and q, the entry of E_(e) at the position (a+(k−1)n,b+(i−1)q) is placed at the position (a, b) of the matrix B_(e).

If e<R, then the software computes E_(e+1)=E_(e)−A_(e)⊗B_(e) and redo sub-Steps 4.1-4.4.

If e=R, then the software performs Step 5.

Step 5: The software collect the matrices found in Step 4 as follows:

N=A ₁ ⊗B ₁ +A ₂ ⊗B ₂ +. . . +A _(R) ⊗B _(R).

The matrix N is the matrix of the output file. If the compression is lossless, then N=M. If the compression is lossy, then N approximates M.

The essential matrix E built at the decompression process can be built at the compression process and replaces the left and right essential matrices A and B. In this case, the compressed file is formed by E and P.

The entries of E are the essential entries of the original matrix M and the non-essential entries can be any number chosen. Thus, using other lossless compression on E will automatically lead to a compressed file that has a memory value less than the memory value of the compressed file if the same lossless compression method is used on M.

In practice, the pattern matrix P can occupy less than 1% of the memory space occupied by M. Thus, compression with BSD has a lossless compression ratio that is more than 0.99. For example, if m=16, n=24, p=8, and q=16, the memory size of the matrix M is 49,152 bytes. Meanwhile, to store the decomposition shape (16, 24, 8, 16) and the pattern matrix P, we need no more than 260 bytes, that is less than 0.53% of the size of M. Thus, the compression ratio is more than 0.994.

In many cases, there are several choices for the parameters m, n, p and q. Some of these choices are more appropriate to adapt to some type of files to be compressed.

Embodiment 5

The second embodiment uses BSD, low rank approximation, and compressing by blocks, to achieve efficient lossy compressions of digital files. To measure the quality of the output file, Peak Signal to Noise Ratio (PSNR) is used.

Assume that the input file is an image represented by a matrix M whose entries are integers in [0, 255], for which we want the entries of the compressed file and output file to be also integers in [0, 255].

For the compression process, the inputs include M, Er, a number that sets a threshold error, and the parameters m, n, p, and q that define the block matrices. We assume that the size of M is several times bigger than the size of an mn×pq-matrix. In the process of compression, M will be split into mn×pq-block matrices.

Step 1: If the height of M is not divisible by mn or the width of M is not divisible by pq, then columns or rows of zeroes are added to M to reach the divisibility. The added parts are then disregarded at the decompressing process.

Step 2: The software splits the image into blocks. Each block is an mn×pq-matrix. Let's say that h is the number of these blocks.

Step 3: For each e=1, . . . ,h, the software performs the following steps for the e^(th) block M_(e).

Step 3.1: Using SVD, the software computes the matrix D_(e) that has the lowest rank possible R_(e) among all matrices for which the PSNR with M is bigger or equal to Er.

Step 3.2: The software rounds up the matrix D_(e) and obtains a matrix Q_(e) whose entries are integers. (Most of the entries of Q_(e) are in [0, 255], but few entries are out of that range).

Step 3.3: Using BSD of M_(e) with R_(e) sets of the steps described in 4.1.2, the software decompresses M_(e) and, as in 4.1.2, builds the left essential, right essential, and pattern matrices of M_(e).

Step 3.4: The software stores the rows of the e^(th) pattern matrix in the rows (2e−1) and 2e of a matrix P, the pattern matrix; the rows of the e^(th) left essential matrix in the rows ((e−1)m+1) to em of a matrix A, the left essential matrix; the rows of the e^(th) right essential matrix in the rows ((e−1)n+1) to en of a matrix B, the right essential matrix, and R_(e) in a sequence of numbers R. The matrices A, B, P, and R along with the size S of the original file constitute the compressed file.

Embodiment 5 Decompression

For the decompression process, the input is the compressed file that includes A, B, P, R, and S.

Step 1. The software extracts, from the input matrices, h, m, n, p, and q.

Step 2. For each e=1, . . . ,h, the apparatus performs the following steps to find the e^(th) block of the output matrix.

Step 2.1. The software extracts from the input matrices, the matrices A_(e), B_(e), and P_(e) and the number R_(e).

Step 2.2. The software uses similar to the decompression process described in 4.1.4 to build a block matrix N_(e) that approximates the block M_(e).

Step 3. The software collects the h blocks built in Step 2 and forms a matrix, and then the software uses S to build N a matrix that approximates M. If we want the entries of the output to be an integer in [0, 255], we may need to quantize N.

Compressing by blocks or tiling allow parallel processing that significantly reduces the compression and decompression time.

Testing

Using Embodiment 5, the inventor has tested the compression and encryption methods of the disclosure. Some of those testing results are as follows.

The test images evaluated are color images in RGB format. Each image is represented by three matrices. Before compressing using BSD, a matrix was formed with the three matrices one next to the other in a horizontal orientation.

To compare the quality of the output images, Peak Signal to Noise Ratio (PSNR) is used which is commonly used to measure the quality of reconstruction of lossy compression. The larger is the value for PSNR, the better is the quality.

SVD can generally achieve a very good PSNR, albeit with the disadvantages described herein, and which are known. The tests show that BSD leads to a PSNR that is at least very close to the PSNR achieved by SVD, while the compression ratio for BSD is significantly better than the compression ratio for SVD.

The first two images “Splash” and “Lenna” are included in the set of test-images of University of South California linked at http://sipi.usc.edu/database/database.php.

The second two images “Deer” and “Spider Web” are part of the set of test-images offered at http://www.imagecompression.info/.

The third two images “kodim01” and “kodim23” are included in Kodak test-images at http://r0k.us/graphics/kodak/.

Each image is compressed using PNG, JPEG, BSD, and SVD. The trial version of PhotoPad Image Editor was used to achieve the PNG compression and the JPEG compression with its highest quality 100. The compressed files with BSD are stored in PNG format.

The output file for SVD is the file that is represented by the matrix Q in the compression process described in Embodiment 5.

The PSNRs for JPEG and BSD are almost the same. The compressed files for SVD and BSD have almost the same number of pixels, but the memory value per pixel for SVD is much larger than the memory value per pixel for BSD.

Images Memory Splash Lenna Deer Spider_Web Kodim01 Kodim23 Value in KB 786.432 786.432 32,032.689 36,363.264 1,179.648 1,179.648 Memory PNG 258 462 20,200 9,370 719 544 Value in KB JPEG 194 221 9,550 4,650 377 259 BSD 47 176 3,040 1,780 373 211 Compression PNG 3.05 1.7 1.58 3.88 1.64 2.12 Ratio JPEG 4.05 3.56 3.35 7.82 3.13 4.55 BSD 16.72 4.47 10.53 20.43 3.16 5.59 PSNR in dB JPEG 35.2605 37.1238 31.0457 45.7255 35.7935 42.9904 BSD 35.4497 37.1894 31.0642 45.7282 35.8746 43.0094 SVD 35.4957 37.3486 31.0935 46.2952 36.0413 43.5003

The difference between the PSNR for SVD and the PSNR for BSD is between 0.03 dB and 0.57 dB. For Splash, Deer, and Spider, the compression ratios for BSD is between 2.6 and 4 times the compression ratios for JPG. For Lenna, kodim01, and kodim23 the compression ratios are slightly improved by BSD vs JPG.

In each of the compression apparatuses of the present invention, other image processing techniques can be implemented to enhance the image quality and lead to further compression. These techniques include entropy encoding, area image compression, and YCbCr conversion.

Embodiment 6

In this embodiment of the disclosure, lossless compression of digital files is achieved using the BSD method of the disclosure. The algorithm used in this embodiment is different from the algorithm used in embodiment 1, but the two algorithms are based on BSD, as disclosed herein, and which can carry out Schmidt decompositions. As explained elsewhere herein, lossy compression can be achieved with a similar method.

Assume that the input file is represented by an mn×pq matrix M that will be compressed using a Schmidt decomposition with respect to the decomposition shape (m, n, p, q). In the course of the compression and decompression processes, we shall define and compute the following matrices and sequence.

PS: The pattern sequence of M.

E: The essential matrix of M, this is an mn×pq matrix.

S: Essential sequence, built using the pigeonhole principle. S shall have two parts S₁ and S₂.

The following sets of steps describe the compression process. We describe the e^(th) set where e is an integer between 1 and the rank of M that will be computed during the compression process. At this set of steps, an mn×pq matrix M_(e) is defined. The compression process starts with M₁=M.

Step 1: The software computes d, the maximum value of the absolute values of the entries of M_(e). Following the lexicographic order, let (r, c) be the position in M_(e) of the first entry d_(e) whose absolute value is equal to d. Position e of the n×q block matrix of M that includes position (r, c) is stored at the e^(th) term of the pattern sequence PS.

Step 2: The e^(th) n×q block matrix of M is stored at the e^(th) n×q block matrix of E; and following the lexicographic order, the entries of the e^(th) n×q block matrix of M are stored in S₁.

Step 3: The software performs the Euclidean division of c by q and find the remainder j. If the remainder is zero, then j is replaced with q. The software then computes i=(c−j)/q+1.

Step 4: The software performs the Euclidean division of r by n and find the remainder l. If the remainder is zero, then l is replaced with n. The software then computes k=(r−l)/n+1.

Step 5: For each integer a between 1 and m and each integer b between 1 and p, the entry of M_(e) at position (l+(a−1)n,j+(b−1)q) is stored at position (a, b) of the matrix A_(e). Following the lexicographic order, if position (l+(a−1)n,j+(b−1)q) of E is unfilled, we store in it the entry of M located at the same position, and we store the same entry in the sequence S₂.

Step 6: For each integer a between 1 and n, and each integer b between 1 and q, the entry of M_(e) at position (a+(k−1)n,b+(i−1)q) is stored at position (a, b) of the matrix B_(e).

Step 7: The software computes the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e). (As detailed elsewhere herein, the disclosure proves the fact that rank (M_(e)−A_(e)⊗B_(e)/d_(e))=rank (M_(e))−1)

If M_(e+1)=0, then the lossless compression of M is achieved.

If M_(e+1)≠0, then Steps 1-7 are performed with M_(e+1).

The process stops after R sets of steps, where R is the Schmidt rank of M.

The lossy compression process includes sets of the same steps 1-6, while the second statement of Step 7 is replaced with a statement that includes an error threshold that stops the process.

The decomposition shape (m, n, p, q), the pattern sequence PS, and the essential sequence S constitute the compressed file, where S is a sequence whose first part is S₁ and second part is S₂. The essential sequence S has R(mp+nq−R) terms extracted from M and PS has R terms, while the matrix M has mpnq entries.

Embodiment 6 Decompression

In the decompression process, the quadruple (m, n, p, q), and the sequences PS and S constitute the input.

Step 1: The apparatus computes R that is equal to the number of terms in PS.

Step 2: The first Rnq terms of S are used to build R n×q block matrices of an mn×pq matrix E (the essential matrix). The positions of the R block matrices are given by PS. The first Rnq terms of S are then deleted from S. The positions of E that are filled after Step 2 are hereafter called essential positions.

Step 3: Starting with e=1 and finishing with e=R, the apparatus performs the following sub-steps. At this step, mn×pq matrices M_(e) are defined. The process starts with M₁=E.

Step 3.1: The apparatus builds the matrix B_(e) whose entries are the entries of the e^(th) n×q block matrix of M_(e). The apparatus computes d, the maximum value of the absolute values of the entries of B_(e). Following the lexicographic order, the apparatus computes (r, c) the position in M_(e) of the first entry d_(e) in B_(e) whose absolute value is equal to d. The entry d_(e) is stored at the e^(th) term of a sequence D.

Step 3.2: The software performs the Euclidean division of c by q and finds the remainder j. If the remainder is zero, then j is replaced with q. The number j is stored at the e^(th) term of a sequence J. The software then computes i=(c−j)/q+1. The number i is stored at the e^(th) term of a sequence I.

Step 3.3: The software performs the Euclidean division of r by n and finds the remainder l. If the remainder is zero, then l is replaced with n. The number l is stored at the e^(th) term of a sequence L. The software then computes k=(r−l)/n+1. The number k is stored at the e^(th) term of a sequence K.

Step 3.4: At this sub-step an an m×p matrix A_(e) is built. For each ordered (a, b), where 1≤a≤m and 1≤b≤p, if (l+(a−1)n,j+(b−1)q) is an essential position, then the position (a, b) of A_(e) is filled with the entry of M_(e) located at position (l+(a−1)n,j+(b−1)q). The other positions of A_(e) can be filled with any value.

Step 3.5: If e=R, the apparatus performs Step 4. If not, the apparatus computes the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e) and performs the sub-steps of Step 3 for M_(e+1).

Step 4: Starting with e=1 and finishing with e=R, the apparatus computes j, the e^(th) term of J and l, the e^(th) term of L. Following the lexicographic order, if (l+(a−1)n,j+(b−1)q) is not an essential position of E, where 1≤a≤m and 1≤b≤p, then this position is filled with the first term of S. This term is then deleted from S and the position (l+(a−1)n,j+(b−1)q) becomes a new essential position of E.

Step 5: At this step, an mn×pq matrix E_(e) is defined, for each e=1, . . . ,R. The process starts with E₁=E. Starting with e=1 and finishing with e=R, the apparatus performs the following sub-steps.

Step 5.1: The apparatus computes i, j, k, and l that are, respectively, the e^(th) term of I, the e^(th) term J, the e^(th) term of K, and the e^(th) term of L.

Step 5.2: At this sub-step, an m×p matrix A_(e) and an n×q matrix B_(e) are built. For each integer a between 1 and m and each integer b between 1 and p, the entry of E_(e) at the position (l+(a−1)n,j+(b−1)q) is placed at position (a, b) of A_(e). The entries of B_(e) are the entries of e^(th) n×q block matrix of E_(e).

If e<R, then an electronic processing apparatus of the disclosure carrying out this method computes E_(e+1)=E_(e)−A_(e)⊗B_(e)/d_(e), where d_(e) is the e^(th) term of D, and performs sub-steps 5.1 and 5.2 for E_(e+1).

If e=R, then the apparatus performs sub-step 5.3.

Step 5.3: The apparatus collects the matrices found in sub-step 5.2 as follows:

N=A ₁ ⊗B ₁ /d ₁ +A ₂ ⊗B ₂ /d ₂ +. . . +A _(R) ⊗B _(R) /d _(R).

The matrix N is the matrix of the output file. If the compression is lossless, then N=M and R is the Schmidt rank of M. If the compression is lossy, then N approximates M and R is less than the Schmidt rank of M.

Additional Examples

In the following additional examples, ‘the apparatus’ refers to an apparatus, including a processor, executing software in accordance with the disclosure.

An example of a computer system which can be used to carry out algorithms described herein is shown and described with respect to FIG. 4, below. Any portion or all of the algorithms can be carried out on one computer, or a digital file or digital data stream to be compressed can be uploaded to a remote server on a LAN or WAN, including a server on the internet. Portions of the algorithms or embodiments herein can be carried out by different computers in the same location, or by computers in different locations. In either case, steps in an algorithm can be carried out by parallel processing, or multiple computers can each execute subroutines of a given algorithm or embodiment.

Using any of the foregoing methods, a server can execute the algorithms of the disclosure upon the digital file or stream provided, which can be an image file, audio file, video file, data file, or any other data source, including streamed data. Once the digital data is compressed, it can be stored in a remote location for later retrieval or forwarding, or can be returned to the address from which the digital file or stream was originally obtained.

Additionally, the compressed file can be sent to an output device, for example a display, printer (including a 3D printer), audio and/or video output device, or any other known or hereinafter developed output device. When a data file or stream has been compressed and/or encrypted in accordance with the disclosure and is returned to the original source, it can then be more efficiently stored at the source, or redistributed to multiple recipients, where significant savings in cost and speed can be realized because the resulting digital file or stream requires substantially fewer bytes for storage and transmission.

Example Computer System

FIG. 4 illustrates the system architecture for a computer system 700, such as a general purpose computer, or a series of interconnected computers, or a computer containing a plurality of processors, on which or with which the disclosure may be implemented. The exemplary computer system of FIG. 4 is for descriptive purposes only. Although the description may refer to terms commonly used in describing particular computer systems, the description and concepts equally apply to other systems, including systems having architectures dissimilar to FIG. 4. Computer system 700 executes software stored on non-transitory media, the software configured to received inputs from other computing devices or from human interface devices.

Computer system 700 includes at least one central processing unit (CPU) 705, or server, which may be implemented with a conventional microprocessor, a random access memory (RAM) 710 for temporary storage of information, and a read only memory (ROM) 715 for permanent storage of information. A memory controller 720 is provided for controlling RAM 710.

A bus 730 interconnects the components of computer system 700. A bus controller 725 is provided for controlling bus 730. An interrupt controller 735 is used for receiving and processing various interrupt signals from the system components.

Mass storage may be provided by DVD ROM 747, or flash or rotating hard disk drive 752, for example. Data and software, including software 400 of the disclosure, may be exchanged with computer system 700 via removable media such as diskette, CD ROM, DVD, Blu Ray, or other optical media 747 connectable to an Optical Media Drive 746 and Controller 745. Alternatively, other media, including for example a media stick, for example a solid state USB drive, may be connected to an External Device Interface 741, and Controller 740. Additionally, another computing device can be connected to computer system 700 through External Device Interface 741, for example by a USB connector, BLUETOOTH connector, Infrared, or WiFi connector, although other modes of connection are known or may be hereinafter developed. A hard disk 752 is part of a fixed disk drive 751 which is connected to bus 730 by controller 750. It should be understood that other storage, peripheral, and computer processing means may be developed in the future, which may advantageously be used with the disclosure.

User input to computer system 700 may be provided by a number of devices. For example, a keyboard 756 and mouse 757 are connected to bus 730 by controller 755. An audio transducer 796, which may act as both a microphone and a speaker, is connected to bus 730 by audio controller 797, as illustrated. It will be obvious to those reasonably skilled in the art that other input devices, such as a pen and/or tablet, Personal Digital Assistant (PDA), mobile/cellular phone and other devices, may be connected to bus 730 and an appropriate controller and software, as required. DMA controller 760 is provided for performing direct memory access to RAM 710. A visual display is generated by video controller 765 which controls video display 770. Computer system 700 also includes a communications adapter 790 which allows the system to be interconnected to a local area network (LAN) or a wide area network (WAN), schematically illustrated by bus 791 and network 795.

Operation of computer system 700 is generally controlled and coordinated by operating system software, such as a Windows system, commercially available from Microsoft Corp., Redmond, Wash. The operating system controls allocation of system resources and performs tasks such as processing scheduling, memory management, networking, and I/O services, among other things. In particular, an operating system resident in system memory and running on CPU 705 coordinates the operation of the other elements of computer system 700. The present disclosure may be implemented with any number of commercially available operating systems.

One or more applications, such as an HTML page server, or a commercially available communication application, may execute under the control of the operating system, operable to convey information to a user.

All references cited herein are expressly incorporated by reference in their entirety. It will be appreciated by persons skilled in the art that the present disclosure is not limited to what has been particularly shown and described herein above. In addition, unless mention was made above to the contrary, it should be noted that all of the accompanying drawings are not to scale. There are many different features to the present disclosure and it is contemplated that these features may be used together or separately. Thus, the disclosure should not be limited to any particular combination of features or to a particular application of the disclosure. Further, it should be understood that variations and modifications within the spirit and scope of the disclosure might occur to those skilled in the art to which the disclosure pertains. Accordingly, all expedient modifications readily attainable by one versed in the art from the disclosure set forth herein that are within the scope and spirit of the present disclosure are to be included as further embodiments of the present disclosure.

REFERENCES

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What is claimed is:
 1. A method for encoding digital data, comprising: using a computer executing software stored on non-transitory media, the software configured to: use SVD to find Ma, an mn×pq matrix, with the lowest Schmidt rank R for which PSNR(Ma, M)≥a predetermined value; quantize Ma to find a matrix Q whose entries are integers; define a left essential matrix A; define a right essential matrix B; define a pattern matrix P for storing positions of essential entries; assign to matrix M_(e) a starting value of Q; define a matrix A_(e); define a matrix B_(e), assign a starting value to e; a) select a non-zero entry d_(e) of M_(e); b) store the position (r, c) of the selected non-zero entry of M_(e) at an e^(th) column of P; c) select from M_(e) two matrices A_(e) and B_(e) having d_(e) as a common entry and for which A_(e)⊗B_(e)/d_(e) is a term in the Schmidt decomposition of M with respect to the parameters m, n, p, and q; d) store in the e^(th) m×p block of A the entries of M whose positions are the positions of the entries of A_(e) in M_(e); e) store in the e^(th) n×q block of B the entries of M whose positions are the positions of the entries of B_(e) in M_(e); and f) calculate the matrix M_(e+1)=M_(e)−A_(e)⊗B_(e)/d_(e), and if e<R , repeat steps (a)-(f) with M_(e+1), and if e=R, then P, A, and B collectively represent encoded digital data corresponding to M.
 2. The method of claim 1, wherein the pattern matrix P is replaced by a sequence of block positions in which the entries of each block matrix are essential entries. 